We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. Show that for every $n \in \mathbf{N}^*$, the random variable $\frac{1+X_n}{2}$ follows a Bernoulli distribution with parameter $\frac{1}{2}$.